20
B. Sutcliffe and R.G. Woolley
The internal molecular Hamiltonian ˆ
H in (1.23) and the clamped-nuclei like operator ˆ
K o just defined can be shown to be essentially self-adjoint (on their respective
Hilbert spaces) by reference to the Kato-Rellich theorem [53] because in both cases
there are kinetic energy operators that dominate the (singular) Coulomb interaction;
they therefore have a complete set of eigenfunctions. As regards ˆ
H elec , we have a
family of Hilbert spaces {H (b)} which are parameterized by the nuclear position
vectors b ∈ X that are the ‘eigenspaces’ of the family of self-adjoint operators ˆ
K o ;
from them we can construct a big Hilbert space as a direct integral over all the b
values
H =
⊕
X
H (b)db
(1.31)
and this is the Hilbert space for ˆ
H elec in (1.25).
Equation (1.31) leads directly to a fundamental result; since ˆ
H elec commutes with
all the {ˆ t
n }, it has the direct integral decomposition
ˆ
H
elec
=
⊕
X
ˆ
K
b, ˆ t
e
o
db.
(1.32)
Even if the ‘ clamped-nuclei’ Hamiltonian has a set of discrete states—Potential Energy Surfaces—(1.32) implies that the unperturbed Hamiltonian, 14 ˆ
H elec , has purely
continuous spectrum (cf. Appendix),
σ = σ
ˆ
H
elec
=
b
σ (b) ≡ [V 0 , ∞)
where V 0 is the minimum value of E(b) 0 ; in the diatomic molecule case this is the
minimum value of the usual ground-state potential energy curve E 0 (r). The operator
ˆ
H elec has no localized eigenfunctions; rather, its eigenfunctions are continuum functions. To avoid any misunderstanding, we emphasize that this result has nothing to
do with the continuous spectrum of the full molecular Hamiltonian associated with
the centre-of-mass motion which can be dealt with trivially in the preliminaries.
A possibly helpful way to think about this paradoxical result is as follows. The
quantum mechanical molecular Hamiltonian for a collection of electrons and nuclei
with Coulomb interactions is a function of position and momentum operators for all
the specified electrons and all the nuclei. If now we separate off the terms containing
all the nuclear momentum operators (the terms proportional to κ 4 ) what is left must
be a function of position and momentum operators for the electrons and position
operators for all the nuclei. This statement is true in any representation of the operators, and in particular must be respected if one chooses a position representation.
This is not what Born and Oppenheimer assumed about their equation (12) [our
equation (1.10)] when κ = 0—see Sect. 1.3.1 above—and which has been assumed
ever since in Quantum Chemistry. In effect they chose to work only in the ‘small’
14 After the elimination of the centre-of-mass variables ˆ
H elec is playing the role of ˆ
H o in (1.20).
B. Sutcliffe and R.G. Woolley
The internal molecular Hamiltonian ˆ
H in (1.23) and the clamped-nuclei like operator ˆ
K o just defined can be shown to be essentially self-adjoint (on their respective
Hilbert spaces) by reference to the Kato-Rellich theorem [53] because in both cases
there are kinetic energy operators that dominate the (singular) Coulomb interaction;
they therefore have a complete set of eigenfunctions. As regards ˆ
H elec , we have a
family of Hilbert spaces {H (b)} which are parameterized by the nuclear position
vectors b ∈ X that are the ‘eigenspaces’ of the family of self-adjoint operators ˆ
K o ;
from them we can construct a big Hilbert space as a direct integral over all the b
values
H =
⊕
X
H (b)db
(1.31)
and this is the Hilbert space for ˆ
H elec in (1.25).
Equation (1.31) leads directly to a fundamental result; since ˆ
H elec commutes with
all the {ˆ t
n }, it has the direct integral decomposition
ˆ
H
elec
=
⊕
X
ˆ
K
b, ˆ t
e
o
db.
(1.32)
Even if the ‘ clamped-nuclei’ Hamiltonian has a set of discrete states—Potential Energy Surfaces—(1.32) implies that the unperturbed Hamiltonian, 14 ˆ
H elec , has purely
continuous spectrum (cf. Appendix),
σ = σ
ˆ
H
elec
=
b
σ (b) ≡ [V 0 , ∞)
where V 0 is the minimum value of E(b) 0 ; in the diatomic molecule case this is the
minimum value of the usual ground-state potential energy curve E 0 (r). The operator
ˆ
H elec has no localized eigenfunctions; rather, its eigenfunctions are continuum functions. To avoid any misunderstanding, we emphasize that this result has nothing to
do with the continuous spectrum of the full molecular Hamiltonian associated with
the centre-of-mass motion which can be dealt with trivially in the preliminaries.
A possibly helpful way to think about this paradoxical result is as follows. The
quantum mechanical molecular Hamiltonian for a collection of electrons and nuclei
with Coulomb interactions is a function of position and momentum operators for all
the specified electrons and all the nuclei. If now we separate off the terms containing
all the nuclear momentum operators (the terms proportional to κ 4 ) what is left must
be a function of position and momentum operators for the electrons and position
operators for all the nuclei. This statement is true in any representation of the operators, and in particular must be respected if one chooses a position representation.
This is not what Born and Oppenheimer assumed about their equation (12) [our
equation (1.10)] when κ = 0—see Sect. 1.3.1 above—and which has been assumed
ever since in Quantum Chemistry. In effect they chose to work only in the ‘small’
14 After the elimination of the centre-of-mass variables ˆ
H elec is playing the role of ˆ
H o in (1.20).